Finding Taylor Polynomial Approximations of Functions - AP Calculus BC

1. State the first four terms of the Taylor series for $\cosh(x)$ centered at $0$.

   $1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \frac{x^6}{6!}$. Even powers only, like cosine but hyperbolic.

2. What is the $2^{nd}$ degree Taylor polynomial for $f(x) = \text{sin}(x)$ centered at $\pi$?

   $0 - (x-\pi)$. $\sin(\pi)=0$, $\sin'(\pi)=-1$ gives linear term.

3. Find the $3^{rd}$ degree Taylor polynomial for $f(x) = \text{ln}(1+x)$ centered at $0$.

   $x - \frac{x^2}{2} + \frac{x^3}{3}$. First three terms of natural log series.

4. What is the $3^{rd}$ degree Taylor polynomial for $f(x) = \text{arcsin}(x)$ at $x = 0$?

   $x + \frac{x^3}{6}$. Arcsine has pattern $\frac{(2n)!x^{2n+1}}{4^n(n!)^2(2n+1)}$.

5. What is the Taylor series for $\text{sin}(x)$ centered at $0$?

   $\text{sin}(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$. Alternating signs with odd powers only.

6. Find the $4^{th}$ degree Taylor polynomial for $f(x) = \text{ln}(x)$ centered at $1$.

   $(x-1) - \frac{(x-1)^2}{2} + \frac{(x-1)^3}{3} - \frac{(x-1)^4}{4}$. Natural log series shifted to center at $x=1$.

7. Identify the convergence condition for a Taylor series.

   Converges if $|x-a| < R$ (radius of convergence $R$). Series converges within its radius of convergence $R$.

8. What is the Taylor series expansion for $\text{ln}(1-x)$ centered at $0$?

   $-x - \frac{x^2}{2} - \frac{x^3}{3} - \cdots$. Replace $x$ with $-x$ in $\ln(1+x)$ series.

9. What is the Taylor series expansion of $e^x$ centered at $0$?

   $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$. Each term is $\frac{x^n}{n!}$ for the exponential function.

10. What is the $1^{st}$ degree Taylor polynomial for $f(x) = \text{e}^x$ centered at $1$?

    $e + e(x-1)$. $e^1=e$ with slope $e$ at $x=1$.

11. What is the remainder term $R_3(x)$ for Taylor polynomial of $f(x) = e^x$ at $x = 0$?

    $\frac{e^c x^4}{4!}$ for some $c \in (0,x)$. Next derivative term after cubic polynomial.

12. What is the general form of the $n^{th}$ degree Taylor polynomial of $f(x)$ centered at $a$?

    $P_n(x) = f(a) + f'(a)(x-a) + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n$. Uses successive derivatives divided by factorials with powers of $(x-a)$.

13. State the Taylor series for $\text{cos}(x)$ centered at $0$.

    $\text{cos}(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$. Alternating signs with even powers only.

14. What is the Taylor polynomial approximation for $f(x) = \text{sin}(x)$ at $x = 0$ up to $x^2$?

    $x$. Only first term survives since $x^2$ term has coefficient zero.

15. Find the Taylor polynomial of degree 2 for $f(x) = x^3$ at $a = 1$.

    $1 + 3(x-1) + 3(x-1)^2$. Taylor expansion of $x^3$ around $x=1$.

16. Find the 3rd degree Taylor polynomial for $f(x) = \text{ln}(1+x)$ centered at $0$.

    $x - \frac{x^2}{2} + \frac{x^3}{3}$. First three terms of natural log series.

17. Determine the Taylor polynomial of degree 1 for $f(x) = \text{ln}(x)$ at $x = 1$.

    $0 + (x-1)$. $f(1)=0$, $f'(1)=1$ gives linear approximation.

18. What is the $4^{th}$ degree Taylor polynomial for $f(x) = x^2 \text{e}^x$ at $x = 0$?

    $0 + 0 + x^2 + x^3 + \frac{x^4}{2}$. Product of $x^2$ and exponential series.

19. Find the Taylor polynomial of degree 2 for $f(x) = \text{e}^{-x}$ at $x = 0$.

    $1 - x + \frac{x^2}{2}$. Replace $x$ with $-x$ in exponential series.

20. Calculate the $3^{rd}$ degree Taylor polynomial for $f(x) = \text{cos}(x)$ at $x = \pi$.

    $-1 + (x-\pi)^2$. $\cos(\pi)=-1$, derivatives give quadratic term.

21. What is the $1^{st}$ degree Taylor polynomial for $f(x) = \text{e}^{2x}$ at $x = 0$?

    $1 + 2x$. Replace $x$ with $2x$ in exponential series.

22. What is the $n^{th}$ degree Taylor polynomial for $f(x) = (1+x)^k$ centered at $0$?

    $1 + kx + \frac{k(k-1)x^2}{2!} + \cdots + \frac{k(k-1)...(k-n+1)x^n}{n!}$. Binomial series with binomial coefficients.

23. Calculate the $2^{nd}$ degree Taylor polynomial for $f(x) = \text{e}^x$ centered at $0$.

    $1 + x + \frac{x^2}{2}$. First three terms of exponential series.

24. What is the $3^{rd}$ degree Taylor polynomial for $f(x) = \text{sin}(2x)$ centered at $0$?

    $2x - \frac{8x^3}{3!}$. Replace $x$ with $2x$ in sine series.

25. State the first four terms of the Taylor series for $\text{cosh}(x)$ centered at $0$.

    $1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \frac{x^6}{6!}$. Even powers only, like cosine but hyperbolic.

26. Determine the Taylor polynomial of degree 1 for $f(x) = \ln(x)$ at $x = 1$.

    $0 + (x-1)$. $f(1)=0$, $f'(1)=1$ gives linear approximation.

27. What is the $1^{st}$ degree Taylor polynomial for $f(x) = e^{2x}$ at $x = 0$?

    $1 + 2x$. Replace $x$ with $2x$ in exponential series.

28. What is the formula for the remainder term $R_n(x)$ in Taylor polynomial?

    $R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}$. Lagrange form with $c$ between $a$ and $x$.

29. Find the first three non-zero terms of the Taylor series for $\text{arctan}(x)$ centered at $0$.

    $x - \frac{x^3}{3} + \frac{x^5}{5}$. Pattern follows $\frac{(-1)^n x^{2n+1}}{2n+1}$.

30. What is the $n^{th}$ degree Taylor polynomial of $f(x) = \text{ln}(1+x)$ centered at $0$?

    $x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots + (-1)^{n-1}\frac{x^n}{n}$. Alternating signs with terms $\frac{(-1)^{n-1}x^n}{n}$.

31. What is the $4^{th}$ degree Taylor polynomial for $f(x) = \text{cos}(x)$ centered at $0$?

    $1 - \frac{x^2}{2} + \frac{x^4}{24}$. Even powers with alternating signs through degree 4.

32. What is the radius of convergence for the Taylor series of $e^x$?

    Radius of convergence is $\infty$. Exponential function converges everywhere.

33. Calculate the $2^{nd}$ degree Taylor polynomial for $f(x) = \text{cos}(3x)$ at $x = 0$.

    $1 - \frac{9x^2}{2}$. Replace $x$ with $3x$ in cosine series.

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